Localization, smoothness, and convergence to equilibrium for a thin film equation

Eric A. Carlen, Süleyman Ulusoy

Research output: Contribution to journalArticle

6 Citations (Scopus)

Abstract

We investigate the long-time behavior of weak solutions to the thin-fllm type equation vt = (xv - vvxxx)x, which arises in the Hele-Shaw problem. We estimate the rate of convergence of solutions to the Smyth-Hill equilibrium solution, which has the form 1/24 (C2 - x 2)2+, in the norm |||f|||2 m,1 =∫R(1 + |x|2m)|f(x)|2 dx + ∫R|f|(x)|2 dx. We obtain exponential convergence in the ||| · |||m,1 norm for all m with 1 ≤ m < 2, thus obtaining rates of convergence in norms measuring both smoothness and localization. The localization is the main novelty, and in fact, we show that there is a close connection between the localization bounds and the smoothness bounds: Convergence of second moments implies convergence in the H1 Sobolev norm. We then use methods of optimal mass transportation to obtain the convergence of the required moments. We also use such methods to construct an appropriate class of weak solutions for which all of the estimates on which our convergence analysis depends may be rigorously derived. Though our main results on convergence can be stated without reference to optimal mass transportation, essential use of this theory is made throughout our analysis.

Original languageEnglish (US)
Pages (from-to)4537-4553
Number of pages17
JournalDiscrete and Continuous Dynamical Systems- Series A
Volume34
Issue number11
DOIs
StatePublished - Nov 2014

Fingerprint

Mass transportation
Thin Film Equation
Convergence to Equilibrium
Smoothness
Norm
Thin films
Weak Solution
Rate of Convergence
Moment
Hele-Shaw
Convergence of Solutions
Exponential Convergence
Equilibrium Solution
Long-time Behavior
Convergence Analysis
Estimate
Imply

All Science Journal Classification (ASJC) codes

  • Analysis
  • Discrete Mathematics and Combinatorics
  • Applied Mathematics

Keywords

  • Euler-Lagrange equation
  • Gradient flow
  • Lyapunov functional
  • Thin-film equation
  • Wasserstein distance

Cite this

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title = "Localization, smoothness, and convergence to equilibrium for a thin film equation",
abstract = "We investigate the long-time behavior of weak solutions to the thin-fllm type equation vt = (xv - vvxxx)x, which arises in the Hele-Shaw problem. We estimate the rate of convergence of solutions to the Smyth-Hill equilibrium solution, which has the form 1/24 (C2 - x 2)2+, in the norm |||f|||2 m,1 =∫R(1 + |x|2m)|f(x)|2 dx + ∫R|f|(x)|2 dx. We obtain exponential convergence in the ||| · |||m,1 norm for all m with 1 ≤ m < 2, thus obtaining rates of convergence in norms measuring both smoothness and localization. The localization is the main novelty, and in fact, we show that there is a close connection between the localization bounds and the smoothness bounds: Convergence of second moments implies convergence in the H1 Sobolev norm. We then use methods of optimal mass transportation to obtain the convergence of the required moments. We also use such methods to construct an appropriate class of weak solutions for which all of the estimates on which our convergence analysis depends may be rigorously derived. Though our main results on convergence can be stated without reference to optimal mass transportation, essential use of this theory is made throughout our analysis.",
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Localization, smoothness, and convergence to equilibrium for a thin film equation. / Carlen, Eric A.; Ulusoy, Süleyman.

In: Discrete and Continuous Dynamical Systems- Series A, Vol. 34, No. 11, 11.2014, p. 4537-4553.

Research output: Contribution to journalArticle

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AU - Carlen, Eric A.

AU - Ulusoy, Süleyman

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AB - We investigate the long-time behavior of weak solutions to the thin-fllm type equation vt = (xv - vvxxx)x, which arises in the Hele-Shaw problem. We estimate the rate of convergence of solutions to the Smyth-Hill equilibrium solution, which has the form 1/24 (C2 - x 2)2+, in the norm |||f|||2 m,1 =∫R(1 + |x|2m)|f(x)|2 dx + ∫R|f|(x)|2 dx. We obtain exponential convergence in the ||| · |||m,1 norm for all m with 1 ≤ m < 2, thus obtaining rates of convergence in norms measuring both smoothness and localization. The localization is the main novelty, and in fact, we show that there is a close connection between the localization bounds and the smoothness bounds: Convergence of second moments implies convergence in the H1 Sobolev norm. We then use methods of optimal mass transportation to obtain the convergence of the required moments. We also use such methods to construct an appropriate class of weak solutions for which all of the estimates on which our convergence analysis depends may be rigorously derived. Though our main results on convergence can be stated without reference to optimal mass transportation, essential use of this theory is made throughout our analysis.

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